An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector w). The mean time until the next occurrence of state s i is [latex] \bar{m}_i= \sum\limits_{k} w_k m_{ki}+ w_i r _i[/latex]. Show that [latex] \bar{m}_i[/latex]= z ii /w i , by using the facts that wZ = w and m ki =(z ii − z ki )/w i .

We know that wZ = w. We also know that m ki = (z ii − z ki )/w i and w i = 1/r i . Putting these in the relation [latex]\bar{m}_i= \sum \limits_k w_km_{ki}+ w_ir_i [/latex] we see that [latex]\bar{m}_i= \sum \limits_k w_k\frac{z_{ii}-z_{ki}}{w_i}+1 \\= \frac{z_{ii}}{w_i}\sum\limits_{k}{w_k}-\frac{1}{w_i}\sum\limits_{k}{w_k z_{ki}}+1 \\= \frac{z_{ii}}{w_i}-1+1 = \frac{z_{ii}}{w_i}.[/latex]

Question 11.5.17: An ergodic Markov chain is started in equilibrium (i.e., wit......

Introduction to Probability – Solutions Manual [1587178]

An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector w). The mean time until the next occurrence of state s_i is $\bar{m}_i= \sum\limits_{k} w_k m_{ki}+ w_i r _i$ . Show that $\bar{m}_i$ = z_ii/w_i, by using the facts that wZ = w and m_ki =(z_ii − z_ki)/w_i.

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